Christoph Glocker, ETH Zurich
Abstract:
Dry friction and impacts lead to discontinuities in the acceleration and the velocity of a mechanical system. Within the framework of Lagrangian mechanics, an algorithm is presented which is able to treat such discontinuity events and to determine the associated state transitions in a most consistent and reliable way. By starting off with Lagrange's equations of second kind, additional contact forces are taken into account by Lagrangian multipliers and are equipped with maximal monotone set-valued constitutive laws, which turn the second order differential equations of the dynamical system into differential inclusions in the sense of Filippov. Generalized Newtonian impact laws in inequality form are consistently embedded into this framework by taking the system's acceleration as the differential measures of its velocity. The resulting measure differential inclusions are discretized on velocity level by an implicit time-stepping scheme. The set-valued constitutive la ws are assumed to be of normal cone type, which relates the problem to optimization theory. For solving the normal cone inclusions, equivalent proximal point problems are formulated and numerically treated by a Gauss-Seidel iteration. This complies with the augmented Lagrangian method used in optimization theory. To demonstrate the power of this approach for structure-variable systems, several applications are presented as e.g. a mechanical snake, an electronic power converter and the analysis of the curve squealing of trains.